search
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
cs-401r:assignment-1 [2014/09/09 14:53] ringger [Question 4: Factoring Joint Distributions] |
cs-401r:assignment-1 [2014/09/24 15:40] (current) cs401rPML [Question 3: Useful theorems in probability theory] added link to example proofs |
||
|---|---|---|---|
| Line 40: | Line 40: | ||
| (Adapted from: Manning & Schuetze, p. 59, exercise 2.1) | (Adapted from: Manning & Schuetze, p. 59, exercise 2.1) | ||
| - | Use the [[Set Theory Identities]] and [[Axioms of Probability Theory]] to prove each of the following five statements. Develop your proof first in terms of sets and then translate into probabilities; use set theoretic operations on sets and arithmetic operators on probabilities. Be sure to apply [[Proofs|good proof technique]]: justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification, you must first satisfy the conditions / pre-requisites of the axiom. | + | Use the [[Set Theory Identities]] and [[Axioms of Probability Theory]] to prove each of the following five statements. Develop your proof first in terms of sets and then translate into probabilities; use set theoretic operations on sets and arithmetic operators on probabilities. Be sure to apply [[Proofs|good proof technique]]: justify each step in your proofs; set up your proofs in two-column format, with each step showing a statement on the left and a justification on the right. Remember that in order to invoke an axiom as justification, you must first satisfy the conditions / pre-requisites of the axiom. See the proofs on the [[example_proofs|example proofs page]]. |
| # $P(B - A) = P(B) - P(A \cap B)$ | # $P(B - A) = P(B) - P(A \cap B)$ | ||
| #* Note that inside the $P(\cdot)$, the '$-$' operator indicates set difference. | #* Note that inside the $P(\cdot)$, the '$-$' operator indicates set difference. | ||
| # $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule) | # $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (the addition rule) | ||
| #* Hint: use the theorem in part #1 as a step in your proof | #* Hint: use the theorem in part #1 as a step in your proof | ||
| - | # $P(\neg A) = 1 - P(A)$ | + | # $P(\overline{A}) = 1 - P(A)$ |
| #* Hint: use the theorem in part #1 as a step in your proof | #* Hint: use the theorem in part #1 as a step in your proof | ||
| - | === Question 4: Factoring Joint Distributions === | + | === Question 4: Factoring Joint Probabilities === |
| [10 points] | [10 points] | ||
| - | # How many possible ways can you completely factor an arbitrary joint probability distribution on six events $P(A_1 \cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$ | + | # How many possible ways can you completely factor the joint probability of six events $P(A_1 \cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$ |
| - | # Given the same joint probability distribution $P(A_1 \cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$, apply the chain rule to completely factor this distribution in one way. | + | # Apply the chain rule to completely factor the joint probability $P(A_1 \cap A_2 \cap A_3 \cap A_4 \cap A_5 \cap A_6)$ in one way. |
| === Question 5: Conditional Probability === | === Question 5: Conditional Probability === | ||
